log, x is always an integer for any choice of real numbers b and x. log, 16 = 4 The collection (0.1.2,3.4) is a group, where the binary " is modular multiplication modulo 5. The collection (1.2.3.4) is a group, where the binary " is modular multiplication modulo 5. OA"group" is any collection of numbers. Zi, is not a group. A generator for the group Z; is: O Any element of the group. An element a E Z; such that a" E Z; for all n E N An element a E Zz; such that Z; = {a, a2,...,a"-1} %3D An element a e Z, such that Z; = {1, a, a2,...,aP-2) Find all the generators of Z*. Write the answer as a sequence of increasing numbers. Separate the numbers with "commas" leave a space after every comma.

Q: 14. Let A = {1,2,3,4,5,6} and p₁ = (4,2,5) and p₂ = (2,6,1) be permutations of A. Compute p₂¹P₁-1…

A:

Q: Let a= (123)(145) in S6. Write a102 in disjoint cycle form. O (32541) O (16)(24) O None of these…

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Q: (12 3 4) o (2 3 5 4 6) o (3 4 5 6) can be expressed as product of following disjoint cycles Select…

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Q: In GF(2), find the inverse of x^3+x+1 modulo x^5+ +x^2+1. Show your complete sölution Oax^3+x…

A: For some finite field  F and fx ,gx∈Fx s.t gx≠0 On applying successive division algorithm:…

Q: Let a= (123)(145) in S6. Write a101 in disjoint cycle form. O None of these (32541) O (14523) O…

A: Given G=S6 α = (123)(145) We have to find  α101 in a disjoint cycle form.

Q: 5. Let 1234 5 67 8 7 4 13 6 5 8 2 p = (a). Write p as a product of disjoint cycles. (2) (b) Write p…

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Q: 7. Suppose that {an) is defined recursively by an = a-1-1 and that ao = 2. Find a3 and 44.

A: Solution :-

Q: 1. Write the following permutations in cycle notation. (1bd from 5.4) (1 2 3 4 5 (1 2 3 4 5 4 3 2 5/…

A: To find- Write the following permutations in cycle notation. 1234542513 1234514325

Q: Please do the following questions with full handwritten working out.

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Q: 1. Build composition table for a dihedral group D4 (Hint: You might need read additional materials…

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Q: Illustrate the Abel's Formula?

A: To  illustrate the Abel's Formula.

Q: One of the one-way functions used in public key cryptography is the discrete logarithm. Computing…

A: We have to find the smallest positive integer e such that r ≡ ge mod p.

Q: (12 3 4) o (2 3 5 4 6) o (3 4 5 6) can be expressed as product of following disjoint cycles Select…

A: To express in disjoint cycles start from right and then go through left to right.

Q: Let a= (123)(145) in S6. Write a102 in disjoint cycle form. (14523) (32541) None of these O (16)(24)

A: See the detailed solution below.

Q: 13. Find another representation 2, in which 3 a. ris negative and 0 < 0 < 2T; b. ris positive and…

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Q: 4.6 List all the elements of (Z45,) that are of order 15.

A: (.) Given  group is  ⊕ .(.). The group  ⊕ is an abelian group under addition modulo   and given by ,…

Q: 9) The 2005 population of the southeast-Aslan country of Bangladesh was approximately 153 million…

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Q: 2. Make a table for each of and ®: 1 3 1 4 1 2 3 4 4 3. 2. 4. 2. 1. 2. 3.

A:

Q: Let a= (123)(145) in S6. Write a101 in disjoint cycle form. O (32541) O (14523) O (16)(24) None of…

A: Given that  a=(1 2 3)(1 4 5)

Q: 4. Convert (101001000)2 from binary to decimal notation and (1984)j10 from decimal to binary…

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Q: (1 2 3 4 5 6 7 \3 4 5 2 7 5 6 E Sz is a product of -------- 2-cycles. a = O 2 Оз 4 O 5

A: .

Q: In GF(2 ), find the multiplicative inverse of (x +x) modulo (x 8+x°+x+x + 1). Use table to show the…

A: Find the Multiplicative inverse of x7+xmodx8+x6+x5+x+1 Without loss, we can verify that for x = 2

Q: 6.4 Find the largest integer smaller than -3/2

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Q: Let a = 1 2 3 4 2 3 5 1 5 6 7 8 478 6 8), B = 1 2 3 4 5 6 7 8 8 4 6 5 1 2, 3 7 (a) Write a, a¹ and ß…

A: Given ,  α= 1234567823514786  ,  β=1234567837846512∈ S8 (.)  Disjoint cycle :  Two cycles are said…

Q: 3. How many zeros at the end does the number 2022! have? (n! is a product of all positive integers…

A: Given,Number = 2022!To find: Number of zeros at the end (trailing zeros) of the given number  As per…

Q: (a) Let a and b be two elements of Sn with the same cycle structure. Explain how to produce an…

A: Good question. Have answered all the details in step 2.

Q: Let a= (123)(145) in S6. Write a202 in disjoint cycle form. (16)(24) (32541) None of these O (14523)

A: Use the concept of permutation group to solve this problem.

Q: Let v= (4,1). Carry out the following computation. Find -7v. |- 7v| = D %3D

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Q: 12. Suppose that {an} is defined recursively by a, = a - 1 that ao = 2. Find az and a4.

A: Given relation is an = an-12 - 1 With a0 = 2

Q: . Write (13)(1245)(13) and (24)(13456)(24) in disjoint c form. Give a simple description of how each…

A: Two permutations in product form are given.

Q: 1. Using the Baby-step Giant-step Algorithm in the group (a) Z, find log, 5, (b) Z7, find log, 11.

A: Baby-step Giant-step Algorithm: Theory : Given a cyclic group G of order p, a generator…

Q: In GF(2), find the inverse of x^3+1 modulo x^5+x^4+x^3+x+1. Show your complete solution. O a.…

A: Check one by one in module 2 and  module the given polynomial

Q: Convert the sum of (ABC19)16 and (19ABC)16 into its binary expansion.

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Q: 6. If *" =x for all x # 0, which of the following must be true? A m-n35 m+n35 m/n-5 mn-5 E mn 5 15.

A:

Q: 1. Define the multiplication operation as modulo 10 in G = {2,4,6.8}. Determine whether the system…

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